3.2.32 \(\int (e x)^m \cos ^p(d (a+b \log (c x^n))) \, dx\) [132]

Optimal. Leaf size=144 \[ \frac {(e x)^{1+m} \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{-p} \cos ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, _2F_1\left (-p,-\frac {i+i m+b d n p}{2 b d n};\frac {1}{2} \left (2-\frac {i (1+m)}{b d n}-p\right );-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (1+m-i b d n p)} \]

[Out]

(e*x)^(1+m)*cos(d*(a+b*ln(c*x^n)))^p*hypergeom([-p, 1/2*(-I-I*m-b*d*n*p)/b/d/n],[1-1/2*I*(1+m)/b/d/n-1/2*p],-e
xp(2*I*a*d)*(c*x^n)^(2*I*b*d))/e/(1+m-I*b*d*n*p)/((1+exp(2*I*a*d)*(c*x^n)^(2*I*b*d))^p)

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Rubi [A]
time = 0.08, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4582, 4580, 371} \begin {gather*} \frac {(e x)^{m+1} \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{-p} \, _2F_1\left (-p,-\frac {i m+b d n p+i}{2 b d n};\frac {1}{2} \left (-\frac {i (m+1)}{b d n}-p+2\right );-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \cos ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (-i b d n p+m+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(e*x)^m*Cos[d*(a + b*Log[c*x^n])]^p,x]

[Out]

((e*x)^(1 + m)*Cos[d*(a + b*Log[c*x^n])]^p*Hypergeometric2F1[-p, -1/2*(I + I*m + b*d*n*p)/(b*d*n), (2 - (I*(1
+ m))/(b*d*n) - p)/2, -(E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))])/(e*(1 + m - I*b*d*n*p)*(1 + E^((2*I)*a*d)*(c*x^n)
^((2*I)*b*d))^p)

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 4580

Int[Cos[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_)*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[Cos[d*(a + b*Log[x])]^p*(x^(
I*b*d*p)/(1 + E^(2*I*a*d)*x^(2*I*b*d))^p), Int[(e*x)^m*((1 + E^(2*I*a*d)*x^(2*I*b*d))^p/x^(I*b*d*p)), x], x] /
; FreeQ[{a, b, d, e, m, p}, x] &&  !IntegerQ[p]

Rule 4582

Int[Cos[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[(e*x)^(m + 1)
/(e*n*(c*x^n)^((m + 1)/n)), Subst[Int[x^((m + 1)/n - 1)*Cos[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a
, b, c, d, e, m, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])

Rubi steps

\begin {align*} \int (e x)^m \cos ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\frac {\left ((e x)^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}}\right ) \text {Subst}\left (\int x^{-1+\frac {1+m}{n}} \cos ^p(d (a+b \log (x))) \, dx,x,c x^n\right )}{e n}\\ &=\frac {\left ((e x)^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}+i b d p} \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{-p} \cos ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right )\right ) \text {Subst}\left (\int x^{-1+\frac {1+m}{n}-i b d p} \left (1+e^{2 i a d} x^{2 i b d}\right )^p \, dx,x,c x^n\right )}{e n}\\ &=\frac {(e x)^{1+m} \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{-p} \cos ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, _2F_1\left (-p,-\frac {i+i m+b d n p}{2 b d n};\frac {1}{2} \left (2-\frac {i (1+m)}{b d n}-p\right );-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (1+m-i b d n p)}\\ \end {align*}

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Mathematica [A]
time = 1.86, size = 170, normalized size = 1.18 \begin {gather*} \frac {x (e x)^m \left (e^{-i a d} \left (c x^n\right )^{-i b d}+e^{i a d} \left (c x^n\right )^{i b d}\right )^p \left (2+2 e^{-2 i a d} \left (c x^n\right )^{-2 i b d}\right )^{-p} \, _2F_1\left (-p,\frac {i (1+m+i b d n p)}{2 b d n};1+\frac {i (1+m)}{2 b d n}-\frac {p}{2};-e^{-2 i a d} \left (c x^n\right )^{-2 i b d}\right )}{1+m+i b d n p} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(e*x)^m*Cos[d*(a + b*Log[c*x^n])]^p,x]

[Out]

(x*(e*x)^m*(1/(E^(I*a*d)*(c*x^n)^(I*b*d)) + E^(I*a*d)*(c*x^n)^(I*b*d))^p*Hypergeometric2F1[-p, ((I/2)*(1 + m +
I*b*d*n*p))/(b*d*n), 1 + ((I/2)*(1 + m))/(b*d*n) - p/2, -(1/(E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)))])/((1 + m +
I*b*d*n*p)*(2 + 2/(E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)))^p)

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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \left (e x \right )^{m} \left (\cos ^{p}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*cos(d*(a+b*ln(c*x^n)))^p,x)

[Out]

int((e*x)^m*cos(d*(a+b*ln(c*x^n)))^p,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*cos(d*(a+b*log(c*x^n)))^p,x, algorithm="maxima")

[Out]

integrate((x*e)^m*cos((b*log(c*x^n) + a)*d)^p, x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*cos(d*(a+b*log(c*x^n)))^p,x, algorithm="fricas")

[Out]

integral((x*e)^m*cos(b*d*log(c*x^n) + a*d)^p, x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**m*cos(d*(a+b*ln(c*x**n)))**p,x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*cos(d*(a+b*log(c*x^n)))^p,x, algorithm="giac")

[Out]

integrate((e*x)^m*cos((b*log(c*x^n) + a)*d)^p, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\cos \left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^p\,{\left (e\,x\right )}^m \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*(a + b*log(c*x^n)))^p*(e*x)^m,x)

[Out]

int(cos(d*(a + b*log(c*x^n)))^p*(e*x)^m, x)

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